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Recently proposed numerical algorithms for solving high-dimensional nonlinearpartial differential equations (PDEs) based on neural networks have shown theirremarkable performance. We review some of them and study their convergenceproperties. The methods rely on probabilistic representation of PDEs bybackward stochastic differential equations (BSDEs) and their iterated timediscretization. Our proposed algorithm, called deep backward multistep scheme(MDBDP), is a machine learning version of the LSMDP scheme of Gobet, Turkedjiev(Math. Comp. 85, 2016). It estimates simultaneously by backward induction thesolution and its gradient by neural networks through sequential minimizationsof suitable quadratic loss functions that are performed by stochastic gradientdescent. Our main theoretical contribution is to provide an approximation erroranalysis of the MDBDP scheme as well as the deep splitting (DS) scheme forsemilinear PDEs designed in Beck, Becker, Cheridito, Jentzen, Neufeld (2019).We also supplement the error analysis of the DBDP scheme of Hur{\'e}, Pham,Warin (Math. Comp. 89, 2020). This yields notably convergence rate in terms ofthe number of neurons for a class of deep Lipschitz continuous GroupSort neuralnetworks when the PDE is linear in the gradient of the solution for the MDBDPscheme, and in the semilinear case for the DBDP scheme. We illustrate ourresults with some numerical tests that are compared with some other machinelearning algorithms in the literature.

Approximation error analysis of some deep backward schemes for nonlinear PDEs